Heat Pump Blower always on

Hi Everyone,

I am having a number of probably related issues with my heat pump. The first issue is that the fan is always on, even with heat off and fan set to auto. The air coming out of the vents is slightly warm, and if left like this the house will heat up. Since the outside system is not making noise, I think this means that the Aux heat is always on, even when the heat is turned off. I should note that when I turn on the heat, the outside unit turns on and the heat pump does work, but when it reaches the set point temperature and the outside unit turns off, the systems keeps blowing relatively warm air.

I have tried two different thermostats. I have an old Honeywell mercury thermostat and a rite temp 6022 programmable tstat.

I have checked the voltages on the thermostat terminals and they seem to make sense. Here is what I find:

I measured with all voltages from C, I'm not sure if that makes sense.

With the heat OFF
With the Fan set to Auto: R is at 24VAC
With the Fan set to ON: R & G are at 24V.
If my reasoning is right, this makes sense, in that G is turning on with the Fan set to on and sending the signal to the HVAC to turn on the fan.
Of course, the Fan is actually running in both these cases.

With the Heat set to ON and the setpoint set below room temperature (i.e. the heat should not be running)
With the Fan set to Auto: R is at 24VAC
With the Fan set to ON: R & G are at 24V.
i.e. same as heat off. makes sense.

With the Heat set to ON and the setpoint set above room temperature (i.e. the heat should running)
The outside unit turns on
With the Fan set to Auto: R, G, Y, and W2 are at 24VAC
With the Fan set to ON: same as above

I should point out that it is cold outside, so it does makes sense that W2 / aux heat would be on.

My results are consistent with both tstats. with the old mercury one it has a light for aux heat that turns on when W2 turns on.

I think everything above makes sense. G is off (with respect to C) when the fan shouldn't be on (i.e. the heat shouldn't be on) and G is on when the fan should be on. Likewise W2 is on when AUX should be and off when it shouldn't. This makes me think that the tstat is fine and the problem is in the unit.

But, as stated above, the fan is always on, and I think the aux is always on too because the air coming out of the vents is slightly warm.

A lot of the forums say that it should be my fan relay. How would I test to figure out if that's the problem? Also, how would I check to figure out if my AUX is actually always on. Is there another explanation for why the air would be warm even though the system is set to off and the outside unit is off?

One last question. my old tstat had an emergency heat position and an E wire. The new thermostat instructions told me not to connect the E wire. Have I lost any functionality?